The reductive GIT quotient \(\pi : Y^{ss} \rightarrow Y/\!/_{\!\mathcal {L}} G\)  is a categorical and good quotient of the open subset \(Y^{ss}:=Y^{G-ss}(\mathcal {L})\)  of GIT semistable points in \(Y\)  (with respect to the \(G\) -linearisation \(\mathcal {L}\) ) and was constructed by Mumford [1]}. The quotient \(Y/\!/_{\!\mathcal {L}} G\)  is a projective scheme equal to the projective spectrum of the graded ring of \(G\) -invariant sections of non-negative powers of \(\mathcal {L}\) . In general, the GIT quotient is not an orbit space: its closed points correspond to S-equivalence classes of semistable orbits, where two semistable orbits are S-equivalent if their orbit closures meet in the semistable set. However, the GIT quotient restricts to a geometric quotient, which in particular is an orbit space, on the open subset \(Y^{s}(\mathcal {L})\subset Y^{ss}\)  of stable points.
